Vanishing viscosity limit for the Navier-Stokes equations
- Series
- PDE Seminar
- Time
- Tuesday, November 30, 2010 - 15:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 255
- Speaker
- Prof. Mikhail Perepelitsa – University of Houston – misha@math.uh.edu
In this talk we will discuss the vanishing viscosity limit of the
Navier-Stokes equations to the isentropic Euler equations for
one-dimensional compressible fluid flow. We will follow the approach of
R.DiPerna (1983) and reduce the problem to the study of a measure-valued
solution of the Euler equations, obtained as a limit of a sequence of
the vanishing viscosity solutions. For a fixed pair (x,t), the (Young)
measure representing the solution encodes the oscillations of the
vanishing viscosity solutions near (x,t). The Tartar-Murat commutator
relation with respect to two pairs of weak entropy-entropy flux kernels
is used to show that the solution takes only Dirac mass values and thus
it is a weak solution of the Euler equations in the usual sense.
In DiPerna's paper and the follow-up papers by other authors this
approach was implemented for the system of the Euler equations with the
artificial viscosity. The extension of this technique to the system of
the Navier-Stokes equations is complicated because of the lack of
uniform (with respect to the vanishing viscosity), pointwise estimates
for the solutions. We will discuss how to obtain the Tartar-Murat
commutator relation and to work out the reduction argument using only
the standard energy estimates.
This is a joint work with Gui-Qiang Chen (Oxford University and Northwestern University).